A359823 Dirichlet inverse of A359820, where A359820 is the characteristic function of numbers whose parity differs from the parity of their arithmetic derivative (A003415).

1, -1, 0, 1, 0, -1, 0, -1, -1, -1, 0, 2, 0, -1, -1, 1, 0, 1, 0, 2, -1, -1, 0, -3, -1, -1, 0, 2, 0, 1, 0, -1, -1, -1, -1, 0, 0, -1, -1, -3, 0, 1, 0, 2, 0, -1, 0, 4, -1, 1, -1, 2, 0, 1, -1, -3, -1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, 2, -1, 1, 0, -2, 0, -1, 0, 2, -1, 1, 0, 4, 0, -1, 0, 1, -1, -1, -1, -3, 0, 3, -1, 2, -1, -1, -1, -5, 0

Offset: 1

Author

Antti Karttunen, Jan 14 2023

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000035, A003415, A359820, A359824 (parity of the terms).

Cf. also A359763 [= a(A003961(n))], A359780.

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A359820(n) = ((n+A003415(n))%2);
memoA359823 = Map();
A359823(n) = if(1==n,1,my(v); if(mapisdefined(memoA359823,n,&v), v, v = -sumdiv(n,d,if(dA359820(n/d)*A359823(d),0)); mapput(memoA359823,n,v); (v)));

Formula

a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, dA359820(n/d) * a(d).